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Non-vanishing of \(L\)-functions and their derivatives. (English) Zbl 0745.11033
Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 89-113 (1990).
[For the entire collection see Zbl 0728.00008.]
The paper in detail discusses the main result of M. R. Murty and V. K. Murty [Ann. Math., II. Ser. 133, No. 3, 447-475 (1991; see the preceding review)] and its implications for modular elliptic curves \(E/\mathbb{Q}\) and also gives an outline of the proofs. In the last section, the author discusses some heuristics on the order of vanishing of \(L(E,s)\) at \(s=1\) and also analogous questions for Artin \(L\)-functions and Dirichlet \(L\)-functions.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions